Borel’s rank theorem for Artin 𝐿-functions

Abstract

Borel’s rank theorem identifies the ranks of algebraic K K -groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin L L -functions by considering equivariant algebraic K K -groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic K K -theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in L L -functions.